A hypergraph $H$ is said to be \emph{linear} if every pair of vertices lies in at most one hyperedge. Given a family $\mathcal{F}$ of $r$-uniform hypergraphs, an $r$-uniform hypergraph $H$ is \emph{$\mathcal{F}$-free} if it contains no member of $\mathcal{F}$ as a subhypergraph. The \emph{linear Turán number} $ex_r^{\mathrm{lin}}(n,\mathcal{F})$ denotes the maximum number of hyperedges in an $\mathcal{F}$-free linear $r$-uniform hypergraph on $n$ vertices. Gyárfás, Ruszinkó, and Sárközy~[\emph{Linear Turán numbers of acyclic triple systems}, European J.\ Combin.\ (2022)] initiated the study of bounds on the linear Turán number for acyclic $3$-uniform linear hypergraphs. In this paper, we extend the study of linear Turán numbers for acyclic systems to higher uniformity. We first give a construction for any linear $r$-uniform tree with $k$ edges that yields the lower bound $ ex_r^{\mathrm{lin}}(n,T_k^r)\ge {n(k-1)}/{r}, $ under mild divisibility and existence assumptions. Next, we study hypertrees with four edges. We prove the exact bound $ ex_r^{\mathrm{lin}}(n,B_4^r)\le {(r+1)n}/{r} $ and characterize the extremal hypergraph class, where $B_4^r$ is formed from $S_3^r$ by appending a hyperedge incident to a degree-one vertex. We also prove the bound $ ex_r^{\mathrm{lin}}(n,E_4^r)\le {(2r-1)n}/{r} $ for the crown $E_4^r$. Finally, we give a construction showing $ ex_r^{\mathrm{lin}}(n,P_4^r)\ge {(r+1)n}/{r} $ under suitable assumptions and conclude with a conjecture on sharp upper bound for $P_4^r$.

翻译：超图 $H$ 称为\textit{线性}的，如果任意两个顶点至多同时出现在一条超边中。给定一个 $r$-一致超图族 $\mathcal{F}$，$r$-一致超图 $H$ 是\textit{无 $\mathcal{F}$ 的}，如果它不包含 $\mathcal{F}$ 中的任何成员作为子超图。\textit{线性Turán数} $ex_r^{\mathrm{lin}}(n,\mathcal{F})$ 表示在 $n$ 个顶点上无 $\mathcal{F}$ 的线性 $r$-一致超图的最大超边数。Gyárfás、Ruszinkó 和 Sárközy~[《无环三元系统的线性Turán数》，European J. Combin. (2022)] 开创了对无环 $3$-一致线性超图的线性Turán数界的研究。本文将无环系统的线性Turán数研究推广到更高一致性。我们首先对任意具有 $k$ 条超边的线性 $r$-一致树给出一个构造，在温和的可整除性和存在性假设下，得到下界 $ ex_r^{\mathrm{lin}}(n,T_k^r)\ge {n(k-1)}/{r} $。接着，我们研究具有四条超边的超树。我们证明了精确界 $ ex_r^{\mathrm{lin}}(n,B_4^r)\le {(r+1)n}/{r} $，并刻画了极值超图类，其中 $B_4^r$ 是通过在 $S_3^r$ 上附加一条与一度顶点关联的超边得到的。对于冠形 $E_4^r$，我们还证明了界 $ ex_r^{\mathrm{lin}}(n,E_4^r)\le {(2r-1)n}/{r} $。最后，我们给出一个构造，在适当假设下表明 $ ex_r^{\mathrm{lin}}(n,P_4^r)\ge {(r+1)n}/{r} $，并得出关于 $P_4^r$ 尖锐上界的猜想。